[Feature]: Let an MCP client label its own extension tab group #43823

Follow-up to #41840 (closed in #43813), re-filed against v1.64.1 with a concrete use case as requested.

What I'm trying to do

I run several AI agent sessions in parallel (Claude Code in a few terminals plus Claude desktop), all driving my everyday Chromium-based browser through the Playwright MCP Bridge extension (--extension). Each session works on a different task, e.g. one checks an order flow, one reads docs, one fills a form.

Since multi-connection support (#42259), each connection gets its own tab group titled Playwright · <clientName> with its own color. That keeps tabs apart, but it doesn't tell me which group belongs to which task:

• All sessions come from the same client, so every group gets the same title (Playwright · claude-code, Playwright · claude-code, ...).
• Colors are assigned in connection order, so they don't map to anything I can remember.
• When a session asks me to finish something by hand (a login, a payment step, a captcha), I have to open each group to find the right tab.

Why existing options don't cover it

clientName comes from the MCP client and is the same for every session of that client. There's no way for the server or the agent to set a more specific group title. Only the agent knows what it's working on, so the label has to be settable at runtime.

Proposal

An MCP tool, available only in --extension mode, that lets a connection label its own tab group:

browser_set_group_label({ label: "checkout flow bug" })

→ the group becomes Playwright · checkout flow bug.

This was already implemented in #42360 (green CI, closed as part of the archiving). The behavior there, plus answers to the open questions from #41840:

• Ownership: a connection can only rename its own group. There's no group/connection id parameter.
• Duplicates: if another active connection uses the same label, a (2), (3) suffix is added.
• Constraints: label is trimmed and capped at a reasonable length. Chrome may still truncate the title in the tab strip, and that's fine.
• Reset: an empty label restores the default Playwright · <clientName>.
• Disconnect: nothing to clean up. The label lives with the connection's group and goes away with it.
• Compatibility: a new session-level extension.setGroupLabel relay command, no protocol bump. An older extension returns a clear "update the extension" error.
• Tests: two connections with the same clientName, each sets a label. Check the titles and that neither can affect the other.

Smaller alternative

If a new tool is too much, a startup option would cover most of my case: --tab-group-label <label> (or an env var) on the MCP server, used instead of clientName in the group title. I'd set it per session when I launch the agent. It's less flexible than the tool, since the label can't follow the task, but it needs no protocol change.

cc Gabriel dos Anjos (@dosxnjos) (author of #41840)